# 2011-2012 Colloquium

This page contains information about the Mathematics Colloquium at the University of Toronto. The colloquium meets regularly on Wednesdays, 4:10-5pm, 6183 Bahen Centre. Refreshments are served before the colloquium in the Math Lounge at 3:30pm. If you will be speaking, details on how to use the equipment in BA 6183 can be found here.

Colloquium announcements are made here: Colloquium announcements

# Fall Term

## September 28, 29, 30, Spielman, 16:10-17:00 in BA1130

 Daniel Spielman [1] (Department of Computer Science Yale University) 2011-2012 Colloquium Wednesday, Thursday, Friday September 28, 29, 30 16:10-17:00 BA1130 Title: The R. A. Blyth Lectures in Mathematics Abstract: Abstracts

## October 5 and 6, Friedman, 15:30 and 11:00 in Fields

 Jerome Friedman (Department of Statistics, Stanford University) 2011-2012 Colloquium Wednesday and Thursday, October 5 and 6 15:30 and 11:00 Fields Title: Fields Distinguished Lecture Series in Statistical Science Abstract: Abstracts

## October 12, Alekseev, 15:10-16:00 in BA6183

 Anton Alekseev (Geneva) 2011-2012 Colloquium Wednesday October 12 15:10-16:00 BA6183 Title: Eigenvalues, interlacing inequalities and tropical calculus Abstract: The same set of inequalities comes up in two seemingly different problems. The first setup is the interlacing inequalities satisfied by the (generalized) eigenvalues of an $n\times n$ Hermitian matrix. By the classical result of Guillemin-Sternberg, they define a completely integrable system (named after Gelfand-Zeitlin who discovered it in the context of Representation Theory of the unitary group). The second setup is the Boltzmann weights associated to (multi-) paths on a planar network with $n$ sources and $n$ sinks. In the talk, we explain the relation between the two theories. As an application, we give a new combinatorial description of the Horn cone (spanned by eigenvalues of triples of Hermitian matrices adding up to zero). This is a joint work with M. Podkopaeva and A. Szenes.

## October 12; 13; 14, Demaine, 16:30-17:30, 11:00-12:00, 11:00-12:00 in SS 2117; FI 230; FI 230

 Eric Demaine (Massachusetts Institute of Technology) Fields Distinguished Lecture Series Wednesday; Thursday; Friday October 12; 13; 14 16:30-17:30, 11:00-12:00, 11:00-12:00 SS 2117; FI 230; FI 230 Title: Algorithms Meet Art, Puzzles, and Magic; Linkage Folding: From Erdös to Proteins; Geometric Puzzles: Algorithms and Complexity Abstract: abstract

## October 19, Morrison, 16:10-17:00 in BA6183

 Scott Morrison http://tqft.net (UC Berkeley) 2011-2012 Colloquium Wednesday October 19 16:10-17:00 BA6183 Title: Fusion categories and subfactors Abstract: Fusion categories provide a simple model for a collection of particles which can split and fuse. Despite the simplicity of the model, it has proved useful in describing exotic materials in condensed matter physics. On the mathematical side, fusion categories can be thought of as 'quantum' finite groups. I'll describe recent joint work on the classification of small fusion categories. The new examples we've encountered are rather strange objects. Moreover, the classification is much sparser than we had expected, encouraging the hope of extending our current knowledge to larger and larger classes of fusion categories. Along the way we'll use analysis (von Neumann algebras and subfactors), number theory (the geometry of cyclotomic integers), graph combinatorics, two-dimensional topology, and some representation theory!

## October 26, TBA, 16:10-17:00 in BA6183 Reserved by

 TBA () 2011-2012 Colloquium Wednesday October 26 16:10-17:00 BA6183 Title: ' Abstract:

## November 2, ', 16:10-17:00 in BA6183 Reserved by

 () 2011-2012 Colloquium Wednesday November 2 16:10-17:00 BA6183 Title: ' Abstract:

## November 9, Seiringer, 16:40-17:30 in BA6183 Reserved by M. Sigal

 Robert Seiringer (McGill University) 2011-2012 Colloquium Wednesday November 9 16:40-17:30 BA6183 Title: Lieb-Thirring Inequalities at Positive Density Abstract: The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schroedinger operator in terms of an L^p norm of the potential. This is dual to a bound on the H^1-norms of a system of orthonormal functions. We present an extension of these inequalities to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials. (This is joint work with R. Frank, M. Lewin and E. Lieb.)

## November 16, TBA, 16:10-17:00 in BA6183 Reserved by

 TBA () 2011-2012 Colloquium Wednesday November 16 16:10-17:00 BA6183 Title: ' Abstract:

## November 23, Kaveh, 16:10-17:00 in BA6183 Reserved by Khovanskii

 Kiumars Kaveh (University of Pittsburgh) 2011-2012 Colloquium Wednesday November 23 16:10-17:00 BA6183 Title: Convex bodies in algebraic geometry and representation theory Abstract: This talk is an invitation to the theory of Okounkov bodies, a new connection between algebraic geometry and convex geometry. It generalizes the well-known and extremely rich correspondence between toric varieties and convex integral polytopes. We will discuss a general combinatorial construction to associate convex bodies to algebraic varieties. It unifies seemingly unrelated constructions in different areas such as Newton polytope of a toric variety, moment polytope of a Hamiltonian action and string polytope of a representation. Among the many applications are elementary proof of the Hodge inequality for intersection numbers, far generalization of the Kushnirenko theorem on the number of solutions of a system of polynomial equation, and construction of (completely) integrable systems on varieties. Following the seminal work of Okounkov, this theory was recently developed and applied in joint work of A. G. Khovanskii and the speaker, and independently by Lazarsfeld and Mustata. Since then its techniques have also been utilized in other areas such as arithmetic geometry, complex geometry and graph coloring. For the most part, the talk is accessible to anyone with just basic background in algebra and geometry.

## November 28, Baum, 16:10-17:00 in BA6183 Reserved for Nov. 28, Elliott

 Paul Baum (Penn State) 2011-2012 Colloquium Monday November 28 16:10-17:00 BA6183 Title: What is K-theory and what is it good for? Abstract: This talk will consist of: 1. The basic definition of K-theory 2. A brief history of K-theory 3. Algebraic versus topological K-theory 4. The unity of K-theory The talk is intended for non-specialists, so the foundational definitions will be carefully stated.

## November 30, Kleiner, 16:10-17:00 in BA6183 Reserved by Kapovitch

 Bruce Kleiner () 2011-2012 Colloquium Wednesday November 30 16:10-17:00 BA6183 Title: Bilipschitz embeddings of metric spaces in Banach spaces Abstract: A map between metric spaces is a bilipschitz homeomorphism if it is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding if it is a bilipschitz homeomorphism onto its image. Given metric spaces X and Y, one may ask if there is a bilipschitz embedding X--->Y, and if so, one may try to find an embedding with minimal distortion, or at least estimate the best bilipschitz constant. Such bilipschitz embedding problems arise in various areas of mathematics, including geometric group theory, Banach space geometry, and geometric analysis; in the last 10 years they have also attracted a lot of attention in theoretical computer science. The lecture will be a survey bilipschitz embedding in Banach spaces from the viewpoint of geometric analysis.

## December 7, Breuillard, 16:10-17:00 in BA6183 CANCELLED

 Emmanuel Breuillard (Université Paris-Sud/MSRI) 2011-2012 Colloquium Wednesday December 7 16:10-17:00 BA6183 Title: Approximate groups Abstract: (CANCELLED)The basic group theory axioms defining a finite group can be relaxed in various natural ways. This leads to the notion of an approximate group, which was introduced a few years ago by Terence Tao in order to tackle the so-called non-commutative inverse Freiman problem: namely to describe the finite subsets A of a given ambient group G such that AA has cardinality at most K times that of A, for some fixed parameter K. In recent years approximate groups have been the focus of a lot of attention as their study lead to a better understanding of the geometric and spectral properties of Cayley graphs of finite simple groups and yielded several applications to number theory (non-commutative sieve methods, spectral gaps for congruence quotients of arithmetic groups, etc). In the talk I will try to give an overview of these developments.

## December 7, Young, 16:10-17:00 in BA6183

 Robert Young (University of Toronto) 2011-2012 Colloquium Wednesday December 7 16:10-17:00 BA6183 Title: Quantifying simple connectivity: an introduction to the Dehn function Abstract: Many theorems start by taking an existence theorem and asking "How many?" or "How big?" or "How fast". The best-known example may be the prime number theorem. Euclid proved that infinitely many primes exist, and the prime number theorem describes how quickly they grow. I'll discuss what happens when you apply the same idea to simple connectivity. In a simply-connected space, any closed curve is the boundary of some disc, but how big is that disc? And what can that tell you about the geometry of the space?

# Spring Term

## January 16, Chaika, 16:10-17:00 in BA6183

 Jon Chaika (University of Chicago) 2011-2012 Colloquium Monday January 16 16:10-17:00 BA6183 Title: TBA Abstract:

## January 18, Siu, 16:10-17:00 in BA6183 Reserved by Murty

 Yum-Tong Siu (Harvard) 2011-2012 Colloquium Wednesday January 18 16:10-17:00 BA6183 Title: Partial Frobenius integrability and its applications Abstract: Will start with the work of Frobenius in 1877 on integrability of distribution of tangent subspaces and trace the development of partial Frobenius integrability through the work of Caratheodory on the second law of thermodynamics in 1909, Chow Wei-liang's generalization in 1939, and Hormander's work on sums of squares of vector fields in 1967. Then will discuss recent applications to Kohn's conjecture on subellipticity for finite-type domains and to the abundance conjecture in algebraic geometry.

## January 20, Zhang, 15:10-16:00 in BA6183

 Zhang () 2011-2012 Colloquium Friday January 20 15:10-16:00 BA6183 Title: TBA Abstract:

## January 23, Roch, 16:10-17:00 in BA6183

 Roch () 2011-2012 Colloquium Monday January 23 16:10-17:00 BA6183 Title: TBA Abstract:

## January 25, Rafi, 16:10-17:00 in BA6183 Reserved by Khanin

 Kasra Rafi (University of Oklahoma) 2011-2012 Colloquium Wednesday January 25 16:10-17:00 BA6183 Title: TBA Abstract:

## January 27, Bejenanu, 15:10-16:00 in BA6183

 Bejenanu () 2011-2012 Colloquium Friday January 27 15:10-16:00 BA6183 Title: TBA Abstract:

## January 30, Schedler, 16:10-17:00 in BA6183 Reserved by Khanin

 Travis Schedler (MIT) 2011-2012 Colloquium Monday January 30 16:10-17:00 BA6183 Title: TBA Abstract:

## February 1, Snyder, 16:10-17:00 in BA6183 Reserved by Khanin

 Noah Snyder (Columbia) 2011-2012 Colloquium Wednesday February 1 16:10-17:00 BA6183 Title: TBA Abstract:

## TBA TBA, Murty, 16:10-17:00 in BA6183 Reserved by Murty

 Ram Murty (Queen's University) 2011-2012 Colloquium TBA TBA TBA 16:10-17:00 BA6183 Title: TBA Abstract:

## February 15, Colding, 16:10-17:00 in BA6183 Reserved by Kapovitch

 Toby Colding (MIT) 2011-2012 Colloquium Wednesday February 15 16:10-17:00 BA6183 Title: Monotonicity formulas revisited and applications Abstract: After discussing various well-known monotonicity formulas in geometric and analysis I will explain several new monotonicity formulas for Ricci curvature and explain some of the things that they can be used for. Among the applications are uniqueness of blow-ups (tangent cones) for Einstein metrics. Uniqueness of blow-ups is a key question in the study of regularity of geometric equation.

## February 29, ', in No colloquium (Coxeter Lecture)

 () Wednesday February 29 Title: ' Abstract:

## March 7, Moore, 16:10-17:00 in BA6183 Reserved by Todorcevic

 Justin Moore (Cornell) 2011-2012 Colloquium Wednesday March 7 16:10-17:00 BA6183 Title: Martin's Maximum Abstract: Since Cohen and Godel's proof that ZFC is incapable of settling the Continuum Hypothesis, there has been an interest in finding additional axioms of set theory which would, at least in practice, settle the truth of mathematical statements left unresolved by ZFC. This has been amplified over the years by a large number of problem arising naturally outside of set theory - in areas such as algebra, analysis, and topology - which turned out to be unresolved by ZFC. I will give an exposition of one axiom which has been extremely successful in settling those mathematical problems left open by ZFC, often in a very satisfying way.

## March 14, Scholze, 16:10-17:00 in Fields Institute, Room 230 Reserved by Herzig

 Peter Scholze (Bonn) 2011-2012 Colloquium Wednesday March 14 16:10-17:00 Fields Institute, Room 230 Title: Perfectoid Spaces Abstract: We will give an informal introduction to the theory of perfectoid spaces, which allows one to compare objects in characteristic p with objects in characteristic 0, with applications e.g. to Deligne's weight-monodromy conjecture.

## March 28, Nekovář, 16:10-17:00 in BA6183 Reserved by Herzig

 Jan Nekovář (Paris 6) 2011-2012 Colloquium Wednesday March 28 16:10-17:00 BA6183 Title: Beyond the Künneth formula Abstract: The Künneth formula states that any reasonable cohomology (with coefficients in a field) of a product X x Y is the tensor product of the cohomology of X with that of Y. There is an important class of complex manifolds (they are, in fact, complex algebraic varieties) which are not products, but whose cohomology is expected to naturally decompose as a direct sum of tensor products. I will try to explain why it would be highly interesting to have a geometric explanation of this phenomenon.

# Colloquium Reservations

• If you wish to reserve a date, email Robert Young (ryoung@math.toronto.edu).